I don't think this is true for all ordinal systems is it? Wouldn't most condorcet methods simply be tallied into a head-to-head table at each precinct and then combined centrally? It's mostly just IRV that has this issue since it's just iterated first past the post, and the order of elimination is important to know in each round before it can be continued. When all that's needed is a h2h table (no iteration) this isn't an issue.
We must note the categorical aspect here. Ordinal is a class of voting, not a specific one. Condorcet is a class of ordinal methods (that optimize the condorcet criteria) but not a specific method. You're right that this isn't true for all methods but I'm mentioning these specific methods because they are the ones that are better and commonly discussed. Unfortunately we can't just look at one aspect of a voting method to rule it in or out.
In essence this is a question of parallelization. Since cardinal systems (minor exception for STAR) your algorithm is argmax(reduce sum(ballots)). Your poll site would report their local reduce sum numbers to the central authority who would reduce sum across all areas and finally perform the argmax operation.
On the other hand, ordinal systems have to be much more coordinated as they are not trivial to parallelize. In essence they are doing an exhaustive search. We run many elections wherein we pop the argmin candidate each round. This has a lock because we cannot pop the argmin candidate from our local polling site until the central authority has collected all ballots and told everyone which candidate to pop.
This is a main motivation for my arguments into cardinal systems. There are other arguments (e.g. favorite betrayer, monotonicity, scalability, preference embedding) but when we are comparing methods and there are minutia to quibble over, algorithmic complexity is a big factor to be concerned with. In fact, I think it is an embarrassingly under discussed feature of extremely high importance. Just imagine how difficult all the recounts in the recent presidential election would have been under a ordinal system (and compare to a cardinal, which the cardinal we can also perform good statistical analysis while we are tallying votes and it is also good for random subset sampling. Also underappreciated features).